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Properties

Label 1008.3.dc.f
Level $1008$
Weight $3$
Character orbit 1008.dc
Analytic conductor $27.466$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,3,Mod(305,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.305");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.dc (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 38 x^{14} - 120 x^{13} + 1059 x^{12} - 3540 x^{11} + 20690 x^{10} - 73200 x^{9} + 269971 x^{8} + \cdots + 352836 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{5} + (\beta_{11} + \beta_{9} + \beta_{4} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{5} + (\beta_{11} + \beta_{9} + \beta_{4} + 1) q^{7} + ( - \beta_{7} - 3 \beta_{3} + 2 \beta_1) q^{11} + (\beta_{12} + \beta_{9} + 3) q^{13} + (\beta_{13} + \beta_{10} + \cdots + 3 \beta_1) q^{17}+ \cdots + ( - 4 \beta_{14} - \beta_{12} + \cdots + 73) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 40 q^{13} + 20 q^{19} - 4 q^{25} + 56 q^{31} + 76 q^{37} - 72 q^{43} - 48 q^{49} - 648 q^{55} - 72 q^{61} + 156 q^{67} + 124 q^{73} - 184 q^{79} - 864 q^{85} + 116 q^{91} + 1112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 38 x^{14} - 120 x^{13} + 1059 x^{12} - 3540 x^{11} + 20690 x^{10} - 73200 x^{9} + 269971 x^{8} + \cdots + 352836 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 61\!\cdots\!55 \nu^{15} + \cdots + 30\!\cdots\!80 ) / 68\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 51\!\cdots\!32 \nu^{15} + \cdots - 40\!\cdots\!60 ) / 20\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 22\!\cdots\!35 \nu^{15} + \cdots + 14\!\cdots\!80 ) / 68\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 98\!\cdots\!30 \nu^{15} + \cdots - 72\!\cdots\!28 ) / 68\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11\!\cdots\!75 \nu^{15} + \cdots + 19\!\cdots\!92 ) / 68\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24\!\cdots\!35 \nu^{15} + \cdots - 66\!\cdots\!00 ) / 10\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 41\!\cdots\!40 \nu^{15} + \cdots + 44\!\cdots\!92 ) / 17\!\cdots\!07 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19\!\cdots\!02 \nu^{15} + \cdots - 63\!\cdots\!40 ) / 68\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 24\!\cdots\!96 \nu^{15} + \cdots - 28\!\cdots\!76 ) / 68\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11\!\cdots\!23 \nu^{15} + \cdots + 67\!\cdots\!76 ) / 20\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 13\!\cdots\!79 \nu^{15} + \cdots + 20\!\cdots\!68 ) / 20\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!03 \nu^{15} + \cdots - 20\!\cdots\!88 ) / 20\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15\!\cdots\!83 \nu^{15} + \cdots - 83\!\cdots\!60 ) / 20\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 19\!\cdots\!96 \nu^{15} + \cdots + 88\!\cdots\!88 ) / 20\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 20\!\cdots\!35 \nu^{15} + \cdots + 38\!\cdots\!00 ) / 10\!\cdots\!42 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{14} - \beta_{12} + \beta_{9} - \beta_{5} - \beta_{4} + 2\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - 4 \beta_{14} + \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 4 \beta_{9} + 2 \beta_{8} + \cdots - 38 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{15} + 20 \beta_{14} + 6 \beta_{13} + 8 \beta_{12} - 20 \beta_{11} - 3 \beta_{10} - 12 \beta_{9} + \cdots + 74 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{14} - 21 \beta_{13} - 32 \beta_{12} + 34 \beta_{11} - 21 \beta_{10} + 32 \beta_{9} + \cdots + 38 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 180 \beta_{15} - 120 \beta_{14} - 10 \beta_{13} + 71 \beta_{12} + 71 \beta_{11} + 95 \beta_{10} + \cdots - 1144 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1079 \beta_{15} + 762 \beta_{14} + 400 \beta_{13} + 624 \beta_{12} - 762 \beta_{11} + 279 \beta_{10} + \cdots + 5896 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 886 \beta_{14} - 3269 \beta_{13} - 3444 \beta_{12} + 2558 \beta_{11} - 3269 \beta_{10} + \cdots + 8771 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 17176 \beta_{15} - 5161 \beta_{14} + 3136 \beta_{13} + 1837 \beta_{12} + 1837 \beta_{11} + 7020 \beta_{10} + \cdots - 48905 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 186111 \beta_{15} + 49600 \beta_{14} + 82449 \beta_{13} + 39604 \beta_{12} - 49600 \beta_{11} + \cdots + 375337 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 24998 \beta_{14} - 286272 \beta_{13} - 79552 \beta_{12} + 54554 \beta_{11} - 286272 \beta_{10} + \cdots + 684139 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 5173245 \beta_{15} - 184866 \beta_{14} + 680735 \beta_{13} - 4516 \beta_{12} - 4516 \beta_{11} + \cdots - 1745893 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 13247608 \beta_{15} - 882678 \beta_{14} + 5675714 \beta_{13} - 763611 \beta_{12} + 882678 \beta_{11} + \cdots - 7225393 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 6856022 \beta_{14} - 75637250 \beta_{13} + 25092474 \beta_{12} - 18236452 \beta_{11} + \cdots + 184626065 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 327162271 \beta_{15} + 76649066 \beta_{14} + 45725167 \beta_{13} - 23121452 \beta_{12} + \cdots + 726497011 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 3149216667 \beta_{15} - 1399952738 \beta_{14} + 1355085003 \beta_{13} - 1049074238 \beta_{12} + \cdots - 9947270231 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-1 - \beta_{4}\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
0.326392 − 1.97954i
−1.92901 + 4.75536i
0.584259 + 0.402247i
1.01836 − 3.17807i
2.24311 − 2.47096i
−0.640485 − 0.304860i
−3.15376 + 4.04825i
1.55114 − 1.27243i
0.326392 + 1.97954i
−1.92901 − 4.75536i
0.584259 − 0.402247i
1.01836 + 3.17807i
2.24311 + 2.47096i
−0.640485 + 0.304860i
−3.15376 − 4.04825i
1.55114 + 1.27243i
0 0 0 −7.42523 + 4.28696i 0 −1.80762 + 6.76258i 0 0 0
305.2 0 0 0 −4.20760 + 2.42926i 0 −6.23005 + 3.19163i 0 0 0
305.3 0 0 0 −0.812456 + 0.469071i 0 1.05091 − 6.92066i 0 0 0
305.4 0 0 0 −0.0443209 + 0.0255887i 0 6.98675 + 0.430549i 0 0 0
305.5 0 0 0 0.0443209 − 0.0255887i 0 6.98675 + 0.430549i 0 0 0
305.6 0 0 0 0.812456 − 0.469071i 0 1.05091 − 6.92066i 0 0 0
305.7 0 0 0 4.20760 − 2.42926i 0 −6.23005 + 3.19163i 0 0 0
305.8 0 0 0 7.42523 − 4.28696i 0 −1.80762 + 6.76258i 0 0 0
737.1 0 0 0 −7.42523 − 4.28696i 0 −1.80762 − 6.76258i 0 0 0
737.2 0 0 0 −4.20760 − 2.42926i 0 −6.23005 − 3.19163i 0 0 0
737.3 0 0 0 −0.812456 − 0.469071i 0 1.05091 + 6.92066i 0 0 0
737.4 0 0 0 −0.0443209 − 0.0255887i 0 6.98675 − 0.430549i 0 0 0
737.5 0 0 0 0.0443209 + 0.0255887i 0 6.98675 − 0.430549i 0 0 0
737.6 0 0 0 0.812456 + 0.469071i 0 1.05091 + 6.92066i 0 0 0
737.7 0 0 0 4.20760 + 2.42926i 0 −6.23005 − 3.19163i 0 0 0
737.8 0 0 0 7.42523 + 4.28696i 0 −1.80762 − 6.76258i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.dc.f 16
3.b odd 2 1 inner 1008.3.dc.f 16
4.b odd 2 1 504.3.cu.b ✓ 16
7.c even 3 1 inner 1008.3.dc.f 16
12.b even 2 1 504.3.cu.b ✓ 16
21.h odd 6 1 inner 1008.3.dc.f 16
28.f even 6 1 3528.3.d.f 8
28.g odd 6 1 504.3.cu.b ✓ 16
28.g odd 6 1 3528.3.d.k 8
84.j odd 6 1 3528.3.d.f 8
84.n even 6 1 504.3.cu.b ✓ 16
84.n even 6 1 3528.3.d.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.3.cu.b ✓ 16 4.b odd 2 1
504.3.cu.b ✓ 16 12.b even 2 1
504.3.cu.b ✓ 16 28.g odd 6 1
504.3.cu.b ✓ 16 84.n even 6 1
1008.3.dc.f 16 1.a even 1 1 trivial
1008.3.dc.f 16 3.b odd 2 1 inner
1008.3.dc.f 16 7.c even 3 1 inner
1008.3.dc.f 16 21.h odd 6 1 inner
3528.3.d.f 8 28.f even 6 1
3528.3.d.f 8 84.j odd 6 1
3528.3.d.k 8 28.g odd 6 1
3528.3.d.k 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{16} - 98 T_{5}^{14} + 7783 T_{5}^{12} - 175394 T_{5}^{10} + 3165901 T_{5}^{8} - 2788988 T_{5}^{6} + \cdots + 16 \) Copy content Toggle raw display
\( T_{11}^{16} - 630 T_{11}^{14} + 258207 T_{11}^{12} - 62644598 T_{11}^{10} + 11074454253 T_{11}^{8} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 98 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{8} + 12 T^{6} + \cdots + 5764801)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{4} - 10 T^{3} + \cdots - 282)^{4} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 489631389843456 \) Copy content Toggle raw display
$19$ \( (T^{8} - 10 T^{7} + \cdots + 14443713124)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 489631389843456 \) Copy content Toggle raw display
$29$ \( (T^{8} + 4286 T^{6} + \cdots + 96387895296)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 1677721782361)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 38 T^{7} + \cdots + 630432824004)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 6076 T^{6} + \cdots + 39080545344)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 18 T^{3} + \cdots + 750694)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 48\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{8} + 36 T^{7} + \cdots + 17348050944)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 1565541478656)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 14533570293264)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 1895820564544)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 386163805404601)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 103199713943076)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{4} - 278 T^{3} + \cdots - 1359456)^{4} \) Copy content Toggle raw display
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